Properties

Label 2-369600-1.1-c1-0-124
Degree $2$
Conductor $369600$
Sign $-1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s − 4·13-s + 2·17-s − 8·19-s + 21-s − 6·23-s − 27-s + 2·31-s + 33-s − 8·37-s + 4·39-s + 10·41-s − 12·43-s − 12·47-s + 49-s − 2·51-s − 2·53-s + 8·57-s − 12·59-s + 2·61-s − 63-s − 8·67-s + 6·69-s − 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.485·17-s − 1.83·19-s + 0.218·21-s − 1.25·23-s − 0.192·27-s + 0.359·31-s + 0.174·33-s − 1.31·37-s + 0.640·39-s + 1.56·41-s − 1.82·43-s − 1.75·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s − 0.125·63-s − 0.977·67-s + 0.722·69-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.66282911777350, −12.25214468785386, −11.97222936821538, −11.35026395670181, −10.94744117271430, −10.30076668194197, −10.13775476067505, −9.758323023979350, −9.132195202347087, −8.602318607638582, −8.108564709495712, −7.702715534736657, −7.159799294913711, −6.655647837121815, −6.189108098498237, −5.903211102307179, −5.199327538628256, −4.709120609060323, −4.412346680271729, −3.722438313531108, −3.168531780532000, −2.581719336850310, −1.900158651174237, −1.579873995098936, −0.4354563198298603, 0, 0.4354563198298603, 1.579873995098936, 1.900158651174237, 2.581719336850310, 3.168531780532000, 3.722438313531108, 4.412346680271729, 4.709120609060323, 5.199327538628256, 5.903211102307179, 6.189108098498237, 6.655647837121815, 7.159799294913711, 7.702715534736657, 8.108564709495712, 8.602318607638582, 9.132195202347087, 9.758323023979350, 10.13775476067505, 10.30076668194197, 10.94744117271430, 11.35026395670181, 11.97222936821538, 12.25214468785386, 12.66282911777350

Graph of the $Z$-function along the critical line